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Defining the logical constants


To: Pat F.
From: Geoffrey Klempner
Subject: Defining the logical constants
Date: 22 November 2007 12:13

Dear Patrick,

Thank you for your email of 10 November, with your University of London Logic essay in response to the question, 'What does it mean to say that an expression is a logical constant?'

Apologies for the delay in my response: on Monday and Tuesday I was in Edinburgh to give a presentation on The metaphysics of the photograph at George Watson's College, Edinburgh. Yesterday, I had to do a drastic shift of gear and prepare unit 2 of my new Pathways Business course on Ethical Dilemmas (for the one student who has so far enrolled on the program).

Today, it's logical constants. Ugh.

Well, Patrick, I checked out the Stanford Encyclopedia (sic) article too (just to brush up on my knowledge, you understand). I don't think you've done too good a job of summarising it. But then, what would be the point.

That aside, I can't help wondering where is the philosophical meat in all of this.

Your (?) idea that 'logical constants can be arbitrarily defined and... the definition then privileges certain logics' does appeal to me because I just can't see this as a real problem. Sure, anything can be a logic.

In the 50's, in the wake of logical positivism, philosophers of religion were writing about 'the logic of God-talk'. It is not necessary to believe (or disbelieve) in the existence of God. Just take the set of statements that you would like to make about God and organize them in the form of a 'logic', which allows certain inferences, disallows others. Then you can say all you want to say about God in the spirit of making moves in a game.

I'm all for this: have as many logics as you like. However, there does come a point where you want to say, 'Look, there are 'logics' and there is LOGIC. LOGIC doesn't need any justification, you can't believe or disbelieve in it. Whereas to use a 'logic' you have to buy into a particular view (maybe a metaphysical view, as in logics of tense and modality) of the world. An axiomatic system set up in the form of a 'logic', with the axioms embedded in its so-called 'logical constants' is just an axiomatic system. It isn't LOGIC.

The next question, however, is, 'How much LOGIC do we need?' Do we need anything apart from propositional and first-order predicate calculus? (assuming it's agreed that Aristotelian logic is inadequate to capture the inferences which we intuitively feel are 'logical').

If the question had been, 'In your opinion, which expressions should count as the logical constants?' my answer would have been, 'The terms used to form propositions in propositional and predicate calculus.' However, the question doesn't ask this. It merely asks, 'What does it mean to say that an expression is a logical constant?' In other words, you are not being asked for your opinion about which are or which are not logical constants (or indeed whether 'it makes no difference what you call a logical constant') but rather, What is at stake? why do we care?

I agree that one acceptable answer would be (as you would like to say) 'Nothing is really at stake. If we call something a logical constant then this follows, and if we don't call it a logical constant then that follows. Take your pick.'

This suggests that the ultimate criterion is indeed pragmatic. We are guided in our decisions by considerations of efficiency, simplicity. There are no metaphysical consequences of our choice, only the consequence that you get to say what you want to say more or less clearly and succinctly.

I suspect that examiners will not be satisfied with this kind of answer. Probably, the best answer to this question would be (as you have probably realized) a well judged distillation of the very thorough Stanford encyclopedia article. Except of course that you have to assume that the examiner has seen the article. So anything that you can read up on this in addition will improve your mark :)

All the best,